Fields Medal Prize Winners (1998)


Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Rewriting Algebraic Fractions
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Rationalizing the Denominator

This document introduces an important method of algebraic simplification used when radicals involve fractions, or equivalently, when a fraction contains radicals in its denominator.

Generally speaking, fractions with radicals in their denominators are considered to be less simple than equivalent fractions with no radicals in their denominator. The process of starting with a fraction containing a radical in its denominator and determining the simplest equivalent fraction with no radical in its denominator is called rationalizing the denominator. (This terminology arises from the fact that generally, radical expressions evaluate to give so-called irrational numbers , whose decimal part never terminates and never achieves a repeated pattern of digits. By eliminating the radical expression from the denominator, we are turning something which gives irrational number values to something which does not give irrational number values – hence the term “rationalizing” the denominator.) There is a general strategy for attempting to rationalize denominators containing radicals of any order. However, in this document, we will focus particularly only on fractions with square roots in the denominator.

Whenever we attempt to manipulate a fraction to get a new fraction which is equivalent but simpler or easier to work with, it is important to ensure that we really do end up with an equivalent fraction. There are many ways to get rid of a square root. (We could simply square it, or we could even just erase it!). We must be careful to use a method which simultaneously makes the square root in the denominator disappear, but leaves us with a mathematically equivalent

To start with one fraction and turn it into another equivalent fraction, we can simply multiply the numerator and denominator by the same thing. Secondly, to rationalize the denominator of a fraction, we could search for some expression that would eliminate all radicals when multiplied onto the denominator. Putting these two observations together, we have a strategy for turning a fraction that has radicals in its denominator into an equivalent fraction with no radicals in the denominator.

(i) deduce a factor which can be multiplied onto the denominator to eliminate all radicals in it. (We will illustrate patterns of such factors for the most commonly occurring simple expressions containing square roots.)

(ii) multiply both the numerator and the denominator of the fraction by this factor. (Then, the radicals will disappear from the denominator, and since we are multiplying the numerator and denominator by the same thing, the result will be a fraction which is equivalent to the original fraction.)

(iii) carry out any necessary or feasible simplifications of the fraction coming out of step (ii).

The tricky part here is step (i). It is easily possible to write down fractions with radicals in their denominators for which step (i) cannot be achieved. However, there are effective standard approaches for most situations commonly occurring in technical applications.


Example 1:



The square root, , in the denominator of this fraction obviously cannot be simplified further. However, it is in the denominator of this fraction, and so to simplify this expression, we do need to rationalize the denominator. To achieve this, we just need to multiply the numerator and denominator of the fraction by the square root occurring in the denominator (since this square root is already in simplest form). This gives

since . The expression on the right just above is a fraction in simplest form. It is equivalent to the original fraction given at the beginning of this example because all we did was multiply the numerator and denominator of that original fraction by the same thing. Finally it has no square root in the denominator. Hence it is the desired final answer here.


People sometimes mistakenly attempt to rationalize the denominator in a fraction such as the one we started with in Example 1 by simply squaring the fraction:

Now, it is true that this final form does have a rational denominator – the square root part has disappeared. However, this final from is in no way equivalent to the original fraction. You can easily see this by noting that

and so, squaring the original fraction amounts here to multiplying the numerator by 5y and the denominator by . Since 5y and are different values except by rare coincidence, this operation does not produce an equivalent fraction. Hence simply squaring a fraction is not an acceptable approach for rationalizing its denominator.